Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves" I hope I'm not distorting his phrase, can someone explain what that means. Lets say that I'm little bit familiar with the ingredients used in both conjectures, Galois representations, elliptic curves,...

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    $\begingroup$ You might look at: math.stackexchange.com/questions/55449/… $\endgroup$ – tttbase May 3 '16 at 12:57
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    $\begingroup$ The answer given by Matthew Emerton in the above linked question probably wont be surpassed here I think. This question is a duplicate of that one, but since the question was asked on the other site, this question cannot be closed for being a duplicate. $\endgroup$ – Stiofán Fordham May 3 '16 at 16:36

The Taniyama conjecture says that the L-series of an elliptic curve over Q is automorphic (more specifically, arises from a modular form). Langlands conjectures that every L-series arising from algebraic geometry is automorphic (in the sense he defined).


To expand on zeno's answer, a Langlands-type formulation of the modularity conjecture would be:

(Taniyama-Shimura) $L(E,s)=L(f,s)$

Here $L(E,s)$ is the Hasse-Weil L-function of an elliptic curve $E$ over $\mathbb{Q}$, and $L(f,s)$ is the L-function of a modular form $f$ of weight 2 with integral coefficients. In fact you have a complete correspondence: all elliptic curves (over $\mathbb{Q}$) are modular in that sense, and all newforms (of that type) come from elliptic curves.

A "Langlands generalization" of this result is:

(Langlands) $L(V,s)=L(\pi,s)$

Here $L(V,s)$ is the Hasse-Weil L-function of any algebraic variety $V$ over an arbitrary number field, and $L(\pi,s)$ the L-function of some automorphic form $\pi$ over $\mathrm{GL}_n(\mathbb{A})$.

This is a theorem in some cases (other than Taniyama-Shimura), for example abelian varieties over $\mathbb{Q}$, elliptic curves over real quadratic fields and elliptic curves with complex multiplication.

You can further generalize:

(Langlands) $L(M,s)=L(\pi,s)$

With $L(M,s)$ the L-function of a motive. This of course includes the case above, Artin L-functions (non-abelian class field theory) and all the other Galois representations that should be automorphic ($l$-adic, mod p).

In the other direction, the Langlands conjectures restricted to the the case of an elliptic curve over $\mathbb{Q}$ predict precisely the same as the Taniyama-Shimura conjecture.


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